3.696 \(\int x^4 (a+b x^2)^{4/3} \, dx\)

Optimal. Leaf size=335 \[ -\frac {432\ 3^{3/4} \sqrt {2-\sqrt {3}} a^4 \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}\right ),4 \sqrt {3}-7\right )}{21505 b^3 x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}-\frac {432 a^3 x \sqrt [3]{a+b x^2}}{21505 b^2}+\frac {48 a^2 x^3 \sqrt [3]{a+b x^2}}{4301 b}+\frac {3}{23} x^5 \left (a+b x^2\right )^{4/3}+\frac {24}{391} a x^5 \sqrt [3]{a+b x^2} \]

[Out]

-432/21505*a^3*x*(b*x^2+a)^(1/3)/b^2+48/4301*a^2*x^3*(b*x^2+a)^(1/3)/b+24/391*a*x^5*(b*x^2+a)^(1/3)+3/23*x^5*(
b*x^2+a)^(4/3)-432/21505*3^(3/4)*a^4*(a^(1/3)-(b*x^2+a)^(1/3))*EllipticF((-(b*x^2+a)^(1/3)+a^(1/3)*(1+3^(1/2))
)/(-(b*x^2+a)^(1/3)+a^(1/3)*(1-3^(1/2))),2*I-I*3^(1/2))*((a^(2/3)+a^(1/3)*(b*x^2+a)^(1/3)+(b*x^2+a)^(2/3))/(-(
b*x^2+a)^(1/3)+a^(1/3)*(1-3^(1/2)))^2)^(1/2)*(1/2*6^(1/2)-1/2*2^(1/2))/b^3/x/(-a^(1/3)*(a^(1/3)-(b*x^2+a)^(1/3
))/(-(b*x^2+a)^(1/3)+a^(1/3)*(1-3^(1/2)))^2)^(1/2)

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Rubi [A]  time = 0.23, antiderivative size = 335, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {279, 321, 236, 219} \[ -\frac {432 a^3 x \sqrt [3]{a+b x^2}}{21505 b^2}-\frac {432\ 3^{3/4} \sqrt {2-\sqrt {3}} a^4 \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right )|-7+4 \sqrt {3}\right )}{21505 b^3 x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}+\frac {48 a^2 x^3 \sqrt [3]{a+b x^2}}{4301 b}+\frac {3}{23} x^5 \left (a+b x^2\right )^{4/3}+\frac {24}{391} a x^5 \sqrt [3]{a+b x^2} \]

Antiderivative was successfully verified.

[In]

Int[x^4*(a + b*x^2)^(4/3),x]

[Out]

(-432*a^3*x*(a + b*x^2)^(1/3))/(21505*b^2) + (48*a^2*x^3*(a + b*x^2)^(1/3))/(4301*b) + (24*a*x^5*(a + b*x^2)^(
1/3))/391 + (3*x^5*(a + b*x^2)^(4/3))/23 - (432*3^(3/4)*Sqrt[2 - Sqrt[3]]*a^4*(a^(1/3) - (a + b*x^2)^(1/3))*Sq
rt[(a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3))^2]*El
lipticF[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3))], -7 +
4*Sqrt[3]])/(21505*b^3*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a + b*x^2)^(
1/3))^2)])

Rule 219

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(2*Sqr
t[2 - Sqrt[3]]*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]*EllipticF[ArcSin[((1 + Sqrt[3
])*s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]])/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[-((s*(s + r*x))/((1 - S
qrt[3])*s + r*x)^2)]), x]] /; FreeQ[{a, b}, x] && NegQ[a]

Rule 236

Int[((a_) + (b_.)*(x_)^2)^(-2/3), x_Symbol] :> Dist[(3*Sqrt[b*x^2])/(2*b*x), Subst[Int[1/Sqrt[-a + x^3], x], x
, (a + b*x^2)^(1/3)], x] /; FreeQ[{a, b}, x]

Rule 279

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
n*p + 1)), x] + Dist[(a*n*p)/(m + n*p + 1), Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x]
&& IGtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rubi steps

\begin {align*} \int x^4 \left (a+b x^2\right )^{4/3} \, dx &=\frac {3}{23} x^5 \left (a+b x^2\right )^{4/3}+\frac {1}{23} (8 a) \int x^4 \sqrt [3]{a+b x^2} \, dx\\ &=\frac {24}{391} a x^5 \sqrt [3]{a+b x^2}+\frac {3}{23} x^5 \left (a+b x^2\right )^{4/3}+\frac {1}{391} \left (16 a^2\right ) \int \frac {x^4}{\left (a+b x^2\right )^{2/3}} \, dx\\ &=\frac {48 a^2 x^3 \sqrt [3]{a+b x^2}}{4301 b}+\frac {24}{391} a x^5 \sqrt [3]{a+b x^2}+\frac {3}{23} x^5 \left (a+b x^2\right )^{4/3}-\frac {\left (144 a^3\right ) \int \frac {x^2}{\left (a+b x^2\right )^{2/3}} \, dx}{4301 b}\\ &=-\frac {432 a^3 x \sqrt [3]{a+b x^2}}{21505 b^2}+\frac {48 a^2 x^3 \sqrt [3]{a+b x^2}}{4301 b}+\frac {24}{391} a x^5 \sqrt [3]{a+b x^2}+\frac {3}{23} x^5 \left (a+b x^2\right )^{4/3}+\frac {\left (432 a^4\right ) \int \frac {1}{\left (a+b x^2\right )^{2/3}} \, dx}{21505 b^2}\\ &=-\frac {432 a^3 x \sqrt [3]{a+b x^2}}{21505 b^2}+\frac {48 a^2 x^3 \sqrt [3]{a+b x^2}}{4301 b}+\frac {24}{391} a x^5 \sqrt [3]{a+b x^2}+\frac {3}{23} x^5 \left (a+b x^2\right )^{4/3}+\frac {\left (648 a^4 \sqrt {b x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a+x^3}} \, dx,x,\sqrt [3]{a+b x^2}\right )}{21505 b^3 x}\\ &=-\frac {432 a^3 x \sqrt [3]{a+b x^2}}{21505 b^2}+\frac {48 a^2 x^3 \sqrt [3]{a+b x^2}}{4301 b}+\frac {24}{391} a x^5 \sqrt [3]{a+b x^2}+\frac {3}{23} x^5 \left (a+b x^2\right )^{4/3}-\frac {432\ 3^{3/4} \sqrt {2-\sqrt {3}} a^4 \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}\right )|-7+4 \sqrt {3}\right )}{21505 b^3 x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}\\ \end {align*}

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Mathematica [C]  time = 0.08, size = 79, normalized size = 0.24 \[ \frac {3 x \sqrt [3]{a+b x^2} \left (\frac {9 a^3 \, _2F_1\left (-\frac {4}{3},\frac {1}{2};\frac {3}{2};-\frac {b x^2}{a}\right )}{\sqrt [3]{\frac {b x^2}{a}+1}}-\left (9 a-17 b x^2\right ) \left (a+b x^2\right )^2\right )}{391 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(a + b*x^2)^(4/3),x]

[Out]

(3*x*(a + b*x^2)^(1/3)*(-((9*a - 17*b*x^2)*(a + b*x^2)^2) + (9*a^3*Hypergeometric2F1[-4/3, 1/2, 3/2, -((b*x^2)
/a)])/(1 + (b*x^2)/a)^(1/3)))/(391*b^2)

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fricas [F]  time = 1.14, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b x^{6} + a x^{4}\right )} {\left (b x^{2} + a\right )}^{\frac {1}{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x^2+a)^(4/3),x, algorithm="fricas")

[Out]

integral((b*x^6 + a*x^4)*(b*x^2 + a)^(1/3), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x^{2} + a\right )}^{\frac {4}{3}} x^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x^2+a)^(4/3),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^(4/3)*x^4, x)

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maple [F]  time = 0.29, size = 0, normalized size = 0.00 \[ \int \left (b \,x^{2}+a \right )^{\frac {4}{3}} x^{4}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(b*x^2+a)^(4/3),x)

[Out]

int(x^4*(b*x^2+a)^(4/3),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x^{2} + a\right )}^{\frac {4}{3}} x^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x^2+a)^(4/3),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^(4/3)*x^4, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x^4\,{\left (b\,x^2+a\right )}^{4/3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a + b*x^2)^(4/3),x)

[Out]

int(x^4*(a + b*x^2)^(4/3), x)

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sympy [A]  time = 1.27, size = 29, normalized size = 0.09 \[ \frac {a^{\frac {4}{3}} x^{5} {{}_{2}F_{1}\left (\begin {matrix} - \frac {4}{3}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(b*x**2+a)**(4/3),x)

[Out]

a**(4/3)*x**5*hyper((-4/3, 5/2), (7/2,), b*x**2*exp_polar(I*pi)/a)/5

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